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S-shaped bifurcation diagrams in exterior domains

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Abstract

We study a nonlinear eigenvalue problem on the exterior to a simply connected bounded domain in \(\mathbb {R}^N\) containing the origin. We consider positive weak solutions satisfying Dirichlet boundary conditions on the compact boundary and decaying to zero at infinity. We discuss multiplicity and uniqueness results of solutions with respect to a bifurcation parameter and conjecture an S-shaped bifurcation diagram for positive reaction terms which are singular at the origin and sublinear at infinity. As a by-product, on regions exterior to a ball with radially symmetric weight functions, we obtain radial symmetry of solutions when uniqueness holds.

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Acknowledgements

The second author was supported by the Grant Agency of the Czech Republic Project No. 18-032523S.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC, 27402, USA

    Maya Chhetri & Ratnasingham Shivaji

  2. Department of Mathematics and NTIS, University of West Bohemia, Univerzitní 8, 30614, Plzeň, Czech Republic

    Pavel Drábek

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  1. Maya Chhetri
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  2. Pavel Drábek
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  3. Ratnasingham Shivaji
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Correspondence to Maya Chhetri.

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Chhetri, M., Drábek, P. & Shivaji, R. S-shaped bifurcation diagrams in exterior domains. Positivity 23, 1147–1164 (2019). https://doi.org/10.1007/s11117-019-00654-8

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